Question

Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(x,y):\mathrm{Aff}/U \to \mathrm{Ens}$ is represented by an algebraic $U$-space, for every $U \in \mathrm{Aff}/S$ and all $x,y \in \mathcal{X}_U$.

I've seen this shown by claiming that the stack associated to $\mathcal{Isom}(x,y)$ is canonically isomorphic to $U \times_{(x,y),\mathcal{X} \times_S \mathcal{X}, \Delta} \mathcal{X}$. My question is how does one make this identification?

Answer 1

This follows immediately from the definitions, in particular from the fiber product of stacks (see loc. cit. (2.2.2)). For $V \in \mathrm{Aff}$ (everything over the base $S$), a $V$-point of $U \times_{X \times X} X$ is a triple $(i,z,\alpha)$, where $i$ is a $V$-point of $U$, $z$ is a $V$-point of $X$ and $\alpha : (i^* x,i^* y) \to (i^*z ,i^* z)$ is an isomorphism of $V$-points of $X \times X$. This comes down to an isomorphism $\alpha_2^{-1} \circ \alpha_1 : i^* x \to i^* y$, i.e. a $V$-point of $\mathrm{Isom}(x,y)$.